Formula-finding device



July 18, 1939. G. EROEDER 2,166,372

FORMULA-FINDING DEVICE Filed June 16, V1938 Patented July 18, 1939 UNITED STATES PATENT OFFICE 3 Claims.

This invention relates to a device for finding formulas for solving triangles and other geometrical figures.

An object of the invention is to find a formula for the solution of a problem, the known and unknown quantities of which are indicated in their proper positions upon a geometrical figure representing the elements of the problem,

Another object of the invention is to make a device in which known and unknown values can be made to appear in their respective positions upon a geometrical figure, and the formula for finding the solution for the unknown values from such known values is simultaneously given.

A further object of my invention is to produce a device in which the values of the elements of a geometrical figure can be made to appear upon said gure While corresponding formulae can be made to appear adjacent said figure.

A still further object of my invention is to provide a device which will quickly and accurately show the method of solution of a problem without the necessity of referring to tables, and/or separated figures, and gives a selection of formula to find an unknown value.

Generally, these objects are obtained by providing two members, one of which is slidable with respect to the other. One of said members has thereon a geometrical figure with apertures, or windows, in the various legs and angles of the figures, and also other windows through which various other information may appear. The second member, which is slidable with respect to the first, has thereon values marked so that when it is displaced with respect to the first member, the various elements of the geometrical figure appear designated by symbols through the windows as known and unknown values. and the formula for ilnding any unknown value from corresponding known values appears through another window. Still other windows simultaneously show the character for the unknown value and an identifying numeral for that particular formula,

The means to achieve these results may be more fully understood by the following description of a device, taken in connection with the accompanying drawing, in which:

Fig. 1 is a plan view of my device.

Fig. 2 is a side view of the device shown in Fig. 1.

Fig. 3 is a plan view of the outer member of the device.

Fig. 4 is a plan view of the inner member of the device.

Fig. 5 is a view showing the inner member partially displaced with respect to the other member to indicate the proper formula for the solution of a problem; and

. Fig. 6 is a view similar to Fig. 5 showing a right triangle.

In Figs. l and 2, the outer member is shown at 2, and the inner member at 4. These members may be made of paper, wood, Celluloid, or any other natural or synthetic material. Outer member 2 has at the transverse edges thereof cut out portions 6 to enable an operators ngers to engage the inner member.

Upon the surface of outer member 2 is printed, or otherwise marked, a geometrical figure 8 which, for purposes of illustration, has been shown to be a triangle. Windows corresponding to the various sides and angles are cut in the outer member, and, as shown, these windows may be cut in the various legs and angles themselves, which windows or apertures extend through the outer member to make visible therethrough the inner member. For example, note Fig. 3, the window Ill designates the upright side of the triangle; the window l2, the horizontal side; the window I3, the inclined side. The angles are also denoted by windows I4, I6, and i8, respectively.

Other windows are also cut which will be briefly referred to and described in detail later. For example, window 20 is cut for the purpose `of showing the formula therethrough from the inner member; Window 22 is cut for denoting the side to be sought; and window 24 is cut for identifying the particular formula being used.

Upon Fig. 4 a plurality of rows of symbols are shown. These symbols are arranged as follows: With the inner member placed coincidently with the outer member as shown in Fig. 1, each leg and angle of the triangle is given a symbol, and these symbols are preferably marked in a distinctv color,

as, for example, a red color. Thus, as shown, the

horizontal side of the triangle is given the symbol A, which symbol A marked upon inner member 4 appears through the window I2 of the outer member 2 as shown in Fig, l. Similarly, the inclined side is given the symbol B; the substantially vertical side, the symbol C; and the angles are given the symbols D, F, and E, all of these being of the same color, preferably red, as the symbol A in this position.

No values appear in the windows 20, 22, and 24 inasmuch as in the position shown in Fig. 1, the various elements of the geometrical gure are merely designated through the windows.

Again referring to Fig. 4, the lowermost row of symbols indicate various formulae for the solution of the triangle for the examples shown when various sides and angles of the triangle are known. This row of formulae is shown at 30, and appears through the window 20 of the outer member. 'Ihe next row 32 shows the symbols A, and certain zero marks which indicate in use that the value of the leg A of the triangle is not known or needed in the solution of the problem. The next row of symbols 34 shows symbols for the angle F. In addition to the letter F and the zero symbols, this row also has certain question marks. These question marks are to 'appear through the window I4 when the angle F is being sought. It is noted that for the Aparticular row, three question marks are shown, and thus three different formulae exist for determiningthe value of the angle F. I

The next row of symbols 36 are for the purpose of showing the values for the angle E through the Window I6, and correspondingly, row 38 shows values for the leg C; row 40, symbols for the leg B through window I3; and row 42,

, symbols for the angle D. Row 44 indicates the unknown quantities, the value of which is sought,

thesesymbols to appear through window v22 in the outer member.` 'For example, the left-hand symbol states that the area 'ofthe' triangle is to be shown; and with reference to Fig. 5, the symbol C indicates'through window `22 that the substantially vertical leg `C lis being sought.

The uppermost row of figures 46 arey to appear through window 24 ofthe outer member,

- and are to be used in connection with the row or the formula desired, can be quickly checked by observing the numeral through the window 24 after the operator of the instrument has learned the association between vthe numerals appearing in row 44, and the formula appearing in row 30.

In Fig. 6, a right triangle is shown, and the values for the side A indicate in row 46' that nve formulae are available for nding side AA.

' It is noted that the Windows on outer member f2 are vertically separated to correspond with the rows of symbols appearing on inner member 4. The device canl be easily ionstructed by rst placing the inner and outer members in a position shown in Fig. 1 andy marking through the windows upon the inner member the symbols designating the various elements of the triangle. The inner member is then displaced with respect to the outer member until the symbols have disappeared, and then an unknown value is marked on either a side or angle by means cfa question mark, preferably in a distinct color, such as red, upon one of the elements of the triangle, and the symbol for said unknown value sought marked through window 22, also in red. The known values are marked in black through the other windows, with the unneeded values marked as zeros through the remaining Windows, and the formula obtaining the unknown values is marked upon the inner member through window 20. As the inner member is successively displaced, other formulae and values are marked upon the inner erably in a black color.

member, and when the inner member iscompletely marked, it takes the appearance shown in Fig. 4.

The above description has been made for the marking of one side of the inner member 4 to be used in connection with the figure delineated upon the surface of the outer member 4. However, the outer member 2 is conveniently made with two sides, 50 and 52, as shown in Fig. 2,

I with the inner member 4 slidable between the two sides.

Therefore, in'such construction, one figure, such as shown by the triangle in Fig. 1, can be marked on one surface of the outer member 2, and, another gure, which may be as shown inFig. 6, or any other geometrical figure, may be marked on the other surface of the outer member 2. Of course, in such a case, the inner member 4 has the .appropriate symbols for such gures placed on each respective side of member 4.

Fig. 5 is an illustration of how the device is used. Thus, in Fig. 5 the operator `seeks to nd a formula by means of which he can determine the value of side C from certain other known values. Consequently, inner member 4 has been vdisplaced with respect to outer member'2 until a question mark appears through Window I0 for side C, and the symbol C appears through window 22, thus showing that C in window 22 cor responds to the side upon whichvthe question -mark appears.l In windowy 24, the numeral 1 appears to show that this is therst formula for determining a value `for leg C. lThe unknown values'appearing through windows I0 and 22 are, as before stated, preferably marked in red. The known values, namely, vthe values for side -A and angles E and D, appear through windows I2, I6,

and I8, respectively, and these symbols are pref- The formula for finding the leg C appears in window 20,'the values for A, D, and E having been given. It thus appears that when it is desired to nd the value of one element of a geometrical figure, -the formula therefor, or a choice of formulae therefor, can be determined by displacing one member with respect to another and having the sought-for value appearas a question mark upon the corresponding element of the geometrical ligure, while a succes/sion of known values can be made to appear in their corresponding places on the other elements of the geometrical ligure, and

This device provides many advantages over prior devices, as, rst of all, the various elements arel made to appear visually lupon the correspondingelements of a geometrical figure, thus enabling the operator to see immediately the relation between the corresponding parts of the geometrical gure in the device, and the elements of the problem which he is attempting to solve. It is further noted that the various question marks or sought-'for values marked upon the rows` of Fig. 4 have been marked successively so that the operator of the device need only move the inner member 4 to two or three successive positions in order to determine all the various formula available for theY solution of that particular unknown value. Through experience, the operator is able to remember that certain formula correspond to the numbers appearing through window 24, and thus, for example, he may remember that a certain formula for finding `the value C corresponds to the numeral l which would appear in window 24, and the operator would then directly slide member 4 until the numerals 1 and C appeared in windows Il and 22, respectively, and nd the formula expected in window Il. .Suchk a correlation between these three numbers provides an additional advantage in that it stimulates the memorization of certain formula by related objects.

Having now described the means by which I obtain the objects of my invention, what I claim is: 4

1. A device for indicating the solution of geometrical. problems comprising an outer member having marked thereon a geometrical ilgure composed of sides and angles, windows arranged in the outlines oi said sides and angles and adapted to disclose iigure indicating indicia, windows located adjacent said gure and adapted to disclose formula and unknown value indicating indicia, an inner member displaceable with respect to said outer member, rows of symbols on said inner member, certain of said symbols in respective rows indicating an unknown value for either a side or an angle through one of said windows arranged in the outlines of said figure, and certain other symbols indicating known values through at least one of the other of the same windows, certain other symbols forming another row of formulae for appearing in said window arranged to disclo formula in dependence upon the symbols appearing in said windows arranged in the outlines o! said ilgure, and other symbols indicative of the side or angle of unknown value appearing in said window arranged to disclose unknown value indicia in dependence upon the appearance of an unknown value symbol through one of said figure windows arranged in the outlines of said gure.

2. A device as in claim 1 further comprising an additional window in said outer member, an additional row of symbols upon said inner member arranged for appearance through said additional window. said additional row of symbols indicating in association with the symbol appearing through said window arranged to disclose unknown value indicia, a particular formula appearing simultaneously through said window arranged to disclose formula.

3. A device for indicating the solution of a geometrical problem comprising a first member having the elements of a geometrical ligure delineated thereon, a plurality of windows cut in said first member, each window corresponding to an element of, and opening through a side or angle of said tlgure, and at least one other window, said windows being vertically displaced with respect to each other across said rst member; n. second member slidable with respect to said rst member, rows of symbols upon said second member corresponding to each of said windows on said rst member, the symbols of certain of said rows indicating through said plurality of windows a character for one of said elements of said gure as either a known or unknown value, the symbols of another row indicating a formula through said other window in dependence upon known value characters simultaneously shown through said plurality of windows, the symbols formed of characters and formulae in said rows presenting through said windows various solutions of geometrical problems relating to said sure.

GEORGE E. ROEDER. 

